Non-asymptotic Convergence of Average-reward Q-learning with Options

This paper proposes a corrected heavy-ball Q-learning method for reinforcement learning (RL) and establishes convergence of its deterministic mean dynamics. It also identifies conditions under which the method is theoretically guaranteed to converge faster than standard Q-learning. The same construction is then extended to Q-learning with linear function approximation, where analogous convergence and acceleration statements are derived for the corresponding corrected fixed point. The sampled stochastic versions are treated through conditional-mean recursions and, in the stated linear-function-approximation setting, finite-time bounds. The analysis is based on a switched linear system (SLS) representation of Q-learning algorithms and on the joint spectral radius (JSR) of the associated switching families. This SLS viewpoint is not commonly used in standard analyses of Q-learning, and it provides a complementary framework and new insight into how heavy-ball momentum can accelerate Q-learning.

In this paper, we derive rates of convergence in the high-dimensional central limit theorem for Polyak-Ruppert averaged iterates generated by the asynchronous Q-learning algorithm with a polynomial stepsize $k^{-ω},\, ω\in (1/2, 1]$. Assuming that the sequence of state-action-next-state triples $(s_k, a_k, s_{k+1})_{k \geq 0}$ forms a uniformly geometrically ergodic Markov chain, we establish a rate of order up to $n^{-1/6} \log^{4} (nS A)$ over the class of hyper-rectangles, where $n$ is the number of samples used by the algorithm and $S$ and $A$ denote the numbers of states and actions, respectively. To obtain this result, we prove a high-dimensional central limit theorem for sums of martingale differences, which may be of independent interest. Finally, we present bounds for high-order moments for the algorithm's last iterate.

The average-reward setting for reinforcement learning is challenging because its Bellman operator is known to contract in span only when the underlying Markov chain is mixing. Whereas existing work typically assumes a contraction property, e.g., through environment mixing, we consider the setting where we are given a reward-agnostic $m$-step mixing policy, and we control the contraction and exploration properties of the learned policy by regularizing toward the mixing policy. We provide finite-time analysis of $Q$-learning algorithms that can be instantiated with multiple choices of regularization, with guarantees measured with respect to the regularized fixed point. Our framework exposes novel regularizer-dependent objects that control contraction, exploration, and operator bias. In the one-step setting, our rates match the best known rates for $Q$-iteration and synchronous $Q$-learning, and for $m$-step mixing we derive new rates. We also study $Q$-planning variants, and show that planning acts as time-localized learning, where the planning horizon $H$ trades off the bias from the boundary value against the sample complexity. Finally, we present numerical studies validating the analysis: (i) knowledge of a strong mixing policy improves convergence and exploration, and its advantage increases with bottleneck severity and, (ii) our theory predicts the convergence and bias rankings of regularizers from the $α$-divergence family.